lsatcvat - Mathbox for Norm Megill

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Theorem lsatcvat 38517

Description: A nonzero subspace less than the sum of two atoms is an atom. (atcvati 32190 analog.) (Contributed by NM, 10-Jan-2015.)

**Hypotheses**
Ref	Expression
lsatcvat.o	⊢ 0 = (0_g‘𝑊)
lsatcvat.s	⊢ 𝑆 = (LSubSp‘𝑊)
lsatcvat.p	⊢ ⊕ = (LSSum‘𝑊)
lsatcvat.a	⊢ 𝐴 = (LSAtoms‘𝑊)
lsatcvat.w	⊢ (𝜑 → 𝑊 ∈ LVec)
lsatcvat.u	⊢ (𝜑 → 𝑈 ∈ 𝑆)
lsatcvat.q	⊢ (𝜑 → 𝑄 ∈ 𝐴)
lsatcvat.r	⊢ (𝜑 → 𝑅 ∈ 𝐴)
lsatcvat.n	⊢ (𝜑 → 𝑈 ≠ { 0 })
lsatcvat.l	⊢ (𝜑 → 𝑈 ⊊ (𝑄 ⊕ 𝑅))

**Assertion**
Ref	Expression
lsatcvat	⊢ (𝜑 → 𝑈 ∈ 𝐴)

**Proof of Theorem lsatcvat**
Step	Hyp	Ref	Expression
1		lsatcvat.o	. . 3 ⊢ 0 = (0_g‘𝑊)
2		lsatcvat.s	. . 3 ⊢ 𝑆 = (LSubSp‘𝑊)
3		lsatcvat.p	. . 3 ⊢ ⊕ = (LSSum‘𝑊)
4		lsatcvat.a	. . 3 ⊢ 𝐴 = (LSAtoms‘𝑊)
5		lsatcvat.w	. . . 4 ⊢ (𝜑 → 𝑊 ∈ LVec)
6	5	adantr 480	. . 3 ⊢ ((𝜑 ∧ ¬ 𝑄 ⊆ 𝑈) → 𝑊 ∈ LVec)
7		lsatcvat.u	. . . 4 ⊢ (𝜑 → 𝑈 ∈ 𝑆)
8	7	adantr 480	. . 3 ⊢ ((𝜑 ∧ ¬ 𝑄 ⊆ 𝑈) → 𝑈 ∈ 𝑆)
9		lsatcvat.q	. . . 4 ⊢ (𝜑 → 𝑄 ∈ 𝐴)
10	9	adantr 480	. . 3 ⊢ ((𝜑 ∧ ¬ 𝑄 ⊆ 𝑈) → 𝑄 ∈ 𝐴)
11		lsatcvat.r	. . . 4 ⊢ (𝜑 → 𝑅 ∈ 𝐴)
12	11	adantr 480	. . 3 ⊢ ((𝜑 ∧ ¬ 𝑄 ⊆ 𝑈) → 𝑅 ∈ 𝐴)
13		lsatcvat.n	. . . 4 ⊢ (𝜑 → 𝑈 ≠ { 0 })
14	13	adantr 480	. . 3 ⊢ ((𝜑 ∧ ¬ 𝑄 ⊆ 𝑈) → 𝑈 ≠ { 0 })
15		lsatcvat.l	. . . 4 ⊢ (𝜑 → 𝑈 ⊊ (𝑄 ⊕ 𝑅))
16	15	adantr 480	. . 3 ⊢ ((𝜑 ∧ ¬ 𝑄 ⊆ 𝑈) → 𝑈 ⊊ (𝑄 ⊕ 𝑅))
17		simpr 484	. . 3 ⊢ ((𝜑 ∧ ¬ 𝑄 ⊆ 𝑈) → ¬ 𝑄 ⊆ 𝑈)
18	1, 2, 3, 4, 6, 8, 10, 12, 14, 16, 17	lsatcvatlem 38516	. 2 ⊢ ((𝜑 ∧ ¬ 𝑄 ⊆ 𝑈) → 𝑈 ∈ 𝐴)
19	5	adantr 480	. . 3 ⊢ ((𝜑 ∧ ¬ 𝑅 ⊆ 𝑈) → 𝑊 ∈ LVec)
20	7	adantr 480	. . 3 ⊢ ((𝜑 ∧ ¬ 𝑅 ⊆ 𝑈) → 𝑈 ∈ 𝑆)
21	11	adantr 480	. . 3 ⊢ ((𝜑 ∧ ¬ 𝑅 ⊆ 𝑈) → 𝑅 ∈ 𝐴)
22	9	adantr 480	. . 3 ⊢ ((𝜑 ∧ ¬ 𝑅 ⊆ 𝑈) → 𝑄 ∈ 𝐴)
23	13	adantr 480	. . 3 ⊢ ((𝜑 ∧ ¬ 𝑅 ⊆ 𝑈) → 𝑈 ≠ { 0 })
24		lveclmod 20985	. . . . . . . . 9 ⊢ (𝑊 ∈ LVec → 𝑊 ∈ LMod)
25	5, 24	syl 17	. . . . . . . 8 ⊢ (𝜑 → 𝑊 ∈ LMod)
26		lmodabl 20786	. . . . . . . 8 ⊢ (𝑊 ∈ LMod → 𝑊 ∈ Abel)
27	25, 26	syl 17	. . . . . . 7 ⊢ (𝜑 → 𝑊 ∈ Abel)
28	2	lsssssubg 20836	. . . . . . . . 9 ⊢ (𝑊 ∈ LMod → 𝑆 ⊆ (SubGrp‘𝑊))
29	25, 28	syl 17	. . . . . . . 8 ⊢ (𝜑 → 𝑆 ⊆ (SubGrp‘𝑊))
30	2, 4, 25, 9	lsatlssel 38464	. . . . . . . 8 ⊢ (𝜑 → 𝑄 ∈ 𝑆)
31	29, 30	sseldd 3980	. . . . . . 7 ⊢ (𝜑 → 𝑄 ∈ (SubGrp‘𝑊))
32	2, 4, 25, 11	lsatlssel 38464	. . . . . . . 8 ⊢ (𝜑 → 𝑅 ∈ 𝑆)
33	29, 32	sseldd 3980	. . . . . . 7 ⊢ (𝜑 → 𝑅 ∈ (SubGrp‘𝑊))
34	3	lsmcom 19807	. . . . . . 7 ⊢ ((𝑊 ∈ Abel ∧ 𝑄 ∈ (SubGrp‘𝑊) ∧ 𝑅 ∈ (SubGrp‘𝑊)) → (𝑄 ⊕ 𝑅) = (𝑅 ⊕ 𝑄))
35	27, 31, 33, 34	syl3anc 1369	. . . . . 6 ⊢ (𝜑 → (𝑄 ⊕ 𝑅) = (𝑅 ⊕ 𝑄))
36	35	psseq2d 4090	. . . . 5 ⊢ (𝜑 → (𝑈 ⊊ (𝑄 ⊕ 𝑅) ↔ 𝑈 ⊊ (𝑅 ⊕ 𝑄)))
37	15, 36	mpbid 231	. . . 4 ⊢ (𝜑 → 𝑈 ⊊ (𝑅 ⊕ 𝑄))
38	37	adantr 480	. . 3 ⊢ ((𝜑 ∧ ¬ 𝑅 ⊆ 𝑈) → 𝑈 ⊊ (𝑅 ⊕ 𝑄))
39		simpr 484	. . 3 ⊢ ((𝜑 ∧ ¬ 𝑅 ⊆ 𝑈) → ¬ 𝑅 ⊆ 𝑈)
40	1, 2, 3, 4, 19, 20, 21, 22, 23, 38, 39	lsatcvatlem 38516	. 2 ⊢ ((𝜑 ∧ ¬ 𝑅 ⊆ 𝑈) → 𝑈 ∈ 𝐴)
41	29, 7	sseldd 3980	. . . . . . 7 ⊢ (𝜑 → 𝑈 ∈ (SubGrp‘𝑊))
42	3	lsmlub 19613	. . . . . . 7 ⊢ ((𝑄 ∈ (SubGrp‘𝑊) ∧ 𝑅 ∈ (SubGrp‘𝑊) ∧ 𝑈 ∈ (SubGrp‘𝑊)) → ((𝑄 ⊆ 𝑈 ∧ 𝑅 ⊆ 𝑈) ↔ (𝑄 ⊕ 𝑅) ⊆ 𝑈))
43	31, 33, 41, 42	syl3anc 1369	. . . . . 6 ⊢ (𝜑 → ((𝑄 ⊆ 𝑈 ∧ 𝑅 ⊆ 𝑈) ↔ (𝑄 ⊕ 𝑅) ⊆ 𝑈))
44		ssnpss 4100	. . . . . 6 ⊢ ((𝑄 ⊕ 𝑅) ⊆ 𝑈 → ¬ 𝑈 ⊊ (𝑄 ⊕ 𝑅))
45	43, 44	syl6bi 253	. . . . 5 ⊢ (𝜑 → ((𝑄 ⊆ 𝑈 ∧ 𝑅 ⊆ 𝑈) → ¬ 𝑈 ⊊ (𝑄 ⊕ 𝑅)))
46	45	con2d 134	. . . 4 ⊢ (𝜑 → (𝑈 ⊊ (𝑄 ⊕ 𝑅) → ¬ (𝑄 ⊆ 𝑈 ∧ 𝑅 ⊆ 𝑈)))
47		ianor 980	. . . 4 ⊢ (¬ (𝑄 ⊆ 𝑈 ∧ 𝑅 ⊆ 𝑈) ↔ (¬ 𝑄 ⊆ 𝑈 ∨ ¬ 𝑅 ⊆ 𝑈))
48	46, 47	imbitrdi 250	. . 3 ⊢ (𝜑 → (𝑈 ⊊ (𝑄 ⊕ 𝑅) → (¬ 𝑄 ⊆ 𝑈 ∨ ¬ 𝑅 ⊆ 𝑈)))
49	15, 48	mpd 15	. 2 ⊢ (𝜑 → (¬ 𝑄 ⊆ 𝑈 ∨ ¬ 𝑅 ⊆ 𝑈))
50	18, 40, 49	mpjaodan 957	1 ⊢ (𝜑 → 𝑈 ∈ 𝐴)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 395 ∨ wo 846 = wceq 1534 ∈ wcel 2099 ≠ wne 2936 ⊆ wss 3945 ⊊ wpss 3946 {csn 4625 ‘cfv 6543 (class class class)co 7415 0_gc0g 17415 SubGrpcsubg 19069 LSSumclsm 19583 Abelcabl 19730 LModclmod 20737 LSubSpclss 20809 LVecclvec 20981 LSAtomsclsa 38441

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-rep 5280 ax-sep 5294 ax-nul 5301 ax-pow 5360 ax-pr 5424 ax-un 7735 ax-cnex 11189 ax-resscn 11190 ax-1cn 11191 ax-icn 11192 ax-addcl 11193 ax-addrcl 11194 ax-mulcl 11195 ax-mulrcl 11196 ax-mulcom 11197 ax-addass 11198 ax-mulass 11199 ax-distr 11200 ax-i2m1 11201 ax-1ne0 11202 ax-1rid 11203 ax-rnegex 11204 ax-rrecex 11205 ax-cnre 11206 ax-pre-lttri 11207 ax-pre-lttrn 11208 ax-pre-ltadd 11209 ax-pre-mulgt0 11210

This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2937 df-nel 3043 df-ral 3058 df-rex 3067 df-rmo 3372 df-reu 3373 df-rab 3429 df-v 3472 df-sbc 3776 df-csb 3891 df-dif 3948 df-un 3950 df-in 3952 df-ss 3962 df-pss 3964 df-nul 4320 df-if 4526 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4905 df-int 4946 df-iun 4994 df-iin 4995 df-br 5144 df-opab 5206 df-mpt 5227 df-tr 5261 df-id 5571 df-eprel 5577 df-po 5585 df-so 5586 df-fr 5628 df-we 5630 df-xp 5679 df-rel 5680 df-cnv 5681 df-co 5682 df-dm 5683 df-rn 5684 df-res 5685 df-ima 5686 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7371 df-ov 7418 df-oprab 7419 df-mpo 7420 df-om 7866 df-1st 7988 df-2nd 7989 df-tpos 8226 df-frecs 8281 df-wrecs 8312 df-recs 8386 df-rdg 8425 df-1o 8481 df-er 8719 df-en 8959 df-dom 8960 df-sdom 8961 df-fin 8962 df-pnf 11275 df-mnf 11276 df-xr 11277 df-ltxr 11278 df-le 11279 df-sub 11471 df-neg 11472 df-nn 12238 df-2 12300 df-3 12301 df-sets 17127 df-slot 17145 df-ndx 17157 df-base 17175 df-ress 17204 df-plusg 17240 df-mulr 17241 df-0g 17417 df-mre 17560 df-mrc 17561 df-acs 17563 df-mgm 18594 df-sgrp 18673 df-mnd 18689 df-submnd 18735 df-grp 18887 df-minusg 18888 df-sbg 18889 df-subg 19072 df-cntz 19262 df-oppg 19291 df-lsm 19585 df-cmn 19731 df-abl 19732 df-mgp 20069 df-rng 20087 df-ur 20116 df-ring 20169 df-oppr 20267 df-dvdsr 20290 df-unit 20291 df-invr 20321 df-drng 20620 df-lmod 20739 df-lss 20810 df-lsp 20850 df-lvec 20982 df-lsatoms 38443 df-lcv 38486

This theorem is referenced by: lsatcvat2 38518

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